Machine Learning and the Axioms of Probability
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Brigham S. Anderson www.cs.cmu.edu/~brigham brigham@cmu.edu School of Computer Science Carnegie Mellon University Probability The world is a very uncertain place 30 years of Artificial Intelligence and Database research danced around this fact And then a few AI researchers decided to use some ideas from the eighteenth century What we’re going to do We will review the fundamentals of probability. It’s really going to be worth it You’ll see examples of probabilistic analytics in action: Inference, Anomaly detection, and Bayes Classifiers Discrete Random Variables A is a Boolean-valued random variable if A denotes an event, and there is some degree of uncertainty as to whether A occurs. Examples A = The US president in 2023 will be male A = You wake up tomorrow with a headache A = You have influenza Probabilities We write P(A) as “the probability that A is true” We could at this point spend 2 hours on the philosophy of this. We’ll spend slightly less... Sample Space Definition 1. The set, S, of all possible outcomes of a particular experiment is called the sample space for the experiment The elements of the sample space are called outcomes . Sample Spaces Sample space of a coin flip: S = {H, T} Sample Spaces Sample space of a die roll: S = {1, 2, 3, 4, 5, 6} Sample Spaces Sample space of three die rolls? S = {111,112,113,…, …,664,665,666} Sample Spaces Sample space of a single draw from a deck of cards: S={As,Ac,Ah,Ad,2s,2c,2h,… …,Ks,Kc,Kd,Kh} So Far… The sample space is the set of all possible worlds. {As,Ac,Ah,Ad,2s,2c,2h,… …,Ks,Kc,Kd,Kh} An outcome is an element of the sample space.
Events Definition 2. An event is any subset of S (including S itself) Events Sample Space of card draw The Sample Space is the set of all outcomes. An Outcome is a possible world. An Event is a set of outcomes Events Sample Space of card draw The Sample Space is the set of all outcomes. An Outcome is a possible world. An Event is a set of outcomes Events Sample Space of card draw The Sample Space is the set of all outcomes. An Outcome is a possible world. An Event is a set of outcomes Definitions The sample space is the set of all possible worlds. {As,Ac,Ah,Ad,2s,2c,2h,… …,Ks,Kc,Kd,Kh} An outcome is a single point in the sample space. An event is a set of outcomes from the sample space.
Events Definition 3. Two events A and B are mutually exclusive if A^B=Ø. Definition 4. If A1, A2, … are mutually exclusive and A1 A2 … = S, then the collection A1, A2, … forms a partition of S. Probability Definition 5. Given a sample space S , a probability function is a function that maps each event in S to a real number, and satisfies P(A) ≥ 0 for any event A in S P(S) = 1 For any number of mutually exclusive events A1 , A2 , A3 …, we have P(A1 A2 A3 …) = P(A1) + P(A2) + P(A3) +… This definition of the domain of this function is not 100% sufficient, but it’s close enough for our purposes… (I’m sparing you Borel Fields) Definitions The sample space is the set of all possible worlds. {As,Ac,Ah,Ad,2s,2c,2h,… …,Ks,Kc,Kd,Kh} An outcome is a single point in the sample space. An event is a set of one or more outcomes P(E) maps event E to a real number and satisfies the axioms of probability P(Red) = 0.50 P(Black) = 0.50
Misconception The relative area of the events determines their probability …in a Venn diagram it does, but not in general. However, the “area equals probability” rule is guaranteed to result in axiom-satisfying probability functions. We often assume, for example, that the probability of “heads” is equal to “tails” in absence of other information… But this is totally outside the axioms! Creating a Valid P() One convenient way to create an axiom-satisfying probability function: Assign a probability to each outcome in S Make sure they sum to one Declare that P(A) equals the sum of outcomes in event A Everyday Example Assume you are a doctor. This is the sample space of “patients you might see on any given day”. Non-smoker, female, diabetic, headache, good insurance, etc… Smoker, male, herniated disk, back pain, mildly schizophrenic, delinquent medical bills, etc… Everyday Example Number of elements in the “patient space”: 100 jillion Are these patients equally likely to occur? Again, generally not. Let’s assume for the moment that they are, though. …which roughly means “area equals probability” Everyday Example Event: Patient has Flu Size of set “F”: 2 jillion (Exactly 2 jillion of the points in the sample space have flu.) Size of “patient space”: 100 jillion Everyday Example From now on, the subscript on P() will be omitted… These Axioms are Not to be Trifled With There have been attempts to do different methodologies for uncertainty Fuzzy Logic Three-valued logic Dempster-Shafer Non-monotonic reasoning But the axioms of probability are the only system with this property: If you gamble using them you can’t be unfairly exploited by an opponent using some other system [di Finetti 1931] Axioms P(A) ≥ 0 for any event A in S P(S) = 1 For any number of mutually exclusive events A1 , A2 , A3 …, we have P(A1 A2 A3 …) = P(A1) + P(A2) + P(A3) +… Theorem. If P is a probability function and A is an event in S, then P(~A) = 1 – P(A) Proof: (1) Since A and ~A partition S, P(A ~A) = P(S) = 1 (2) Since A and ~A are disjoint, P(A ~A) = P(A) + P(~A) Combining (1) and (2) gives the result Multivalued Random Variables Suppose A can take on more than 2 values A is a random variable with arity k if it can take on exactly one value out of {A1,A2, ... Ak}, and The events {A1,A2,…,Ak} partition S, so Elementary Probability in Pictures Elementary Probability in Pictures P(B) = P(B, A) + P(B, ~A) Elementary Probability in Pictures Elementary Probability in Pictures Conditional Probability Assume once more that you are a doctor. Again, this is the sample space of “patients you might see on any given day”. Conditional Probability Conditional Probability Event: Headache P(H) = 0.10 Conditional Probability P(F) = 0.02 P(H) = 0.10 …we still need to specify the interaction between flu and headache… Define P(H|F) = Fraction of F’s outcomes which are also in H Conditional Probability P(F) = 0.02 P(H) = 0.10 P(H|F) = 0.50 Conditional Probability P(H|F) = Fraction of flu worlds in which patient has a headache = #worlds with flu and headache ------------------------------------ #worlds with flu = Size of “H and F” region ------------------------------ Size of “F” region = P(H, F) ---------- P(F) Conditional Probability Definition. If A and B are events in S, and P(B) > 0, then the conditional probability of A given B, written P(A|B), is The Chain Rule A simple rearrangement of the above equation yields Probabilistic Inference H = “Have a headache” F = “Coming down with Flu” P(H) = 0.10 P(F) = 0.02 P(H|F) = 0.50 One day you wake up with a headache. You think: “Drat! 50% of flus are associated with headaches so I must have a 50-50 chance of coming down with flu” Is this reasoning good? Probabilistic Inference H = “Have a headache” F = “Coming down with Flu” P(H) = 0.10 P(F) = 0.02 P(H|F) = 0.50 What we just did… P(A,B) P(A|B) P(B) P(B|A) = ----------- = --------------- P(A) P(A) This is Bayes Rule Bayes, Thomas (1763) An essay towards solving a problem in the doctrine of chances. Philosophical Transactions of the Royal Society of London, 53:370-418 More General Forms of Bayes Rule More General Forms of Bayes Rule Independence Definition. Two events, A and B, are statistically independent if Representing P(A,B,C) How can we represent the function P(A)? P(A,B)? P(A,B,C)? The Joint Probability Table Recipe for making a joint distribution of M variables: Make a truth table listing all combinations of values of your variables (if there are M boolean variables then the table will have 2M rows). For each combination of values, say how probable it is. If you subscribe to the axioms of probability , those numbers must sum to 1. Using the Joint One you have the JPT you can ask for the probability of any logical expression Using the Joint Using the Joint Inference with the Joint Inference with the Joint P(Male | Poor) = 0.4654 / 0.7604 = 0.612 Inference is a big deal I’ve got this evidence. What’s the chance that this conclusion is true? I’ve got a sore neck: how likely am I to have meningitis? I see my lights are out and it’s 9pm. What’s the chance my spouse is already asleep? There’s a thriving set of industries growing based around Bayesian Inference. Highlights are: Medicine, Pharma, Help Desk Support, Engine Fault Diagnosis Where do Joint Distributions come from? Idea One: Expert Humans Idea Two: Simpler probabilistic facts and some algebra Example: Suppose you knew P(A) = 0.5 P(B|A) = 0.2 P(B|~A) = 0.1 P(C|A,B) = 0.1 P(C|A,~B) = 0.8 P(C|~A,B) = 0.3 P(C|~A,~B) = 0.1 Then you can automatically compute the JPT using the chain rule P(A,B,C) = P(A) P(B|A) P(C|A,B) Bayes Nets are a systematic way to do this. Where do Joint Distributions come from? Idea Three: Learn them from data! Prepare to witness an impressive learning algorithm…. Learning a JPT Then fill in each row with Fraction of all records in which A and B are True but C is False Example of Learning a JPT This JPT was obtained by learning from three attributes in the UCI “Adult” Census Database [Kohavi 1995] Where are we? We have recalled the fundamentals of probability We have become content with what JPTs are and how to use them And we even know how to learn JPTs from data. Density Estimation Our Joint Probability Table (JPT) learner is our first example of something called Density Estimation A Density Estimator learns a mapping from a set of attributes to a Probability Evaluating a density estimator Given a record x , a density estimator M can tell you how likely the record is: Given a dataset with R records, a density estimator can tell you how likely the dataset is: (Under the assumption that all records were independently generated from the probability function) A small dataset: Miles Per Gallon From the UCI repository (thanks to Ross Quinlan) A small dataset: Miles Per Gallon A small dataset: Miles Per Gallon Log Probabilities Since probabilities of datasets get so small we usually use log probabilities A small dataset: Miles Per Gallon Summary: The Good News The JPT allows us to learn P(X) from data. Can do inference: P(E1|E2) Automatic Doctor, Recommender, etc Can do anomaly detection spot suspicious / incorrect records (e.g., credit card fraud) Can do Bayesian classification Predict the class of a record (e.g, predict cancerous/not-cancerous) Density estimation with JPTs is trivial, mindless and dangerous Using a test set An independent test set with 196 cars has a much worse log likelihood than it had on the training set (actually it’s a billion quintillion quintillion quintillion quintillion times less likely) ….Density estimators can overfit. And the JPT estimator is the overfittiest of them all! Overfitting Density Estimators If this ever happens, it means there are certain combinations that we learn are “impossible” Using a test set The only reason that our test set didn’t score -infinity is that Andrew’s code is hard-wired to always predict a probability of at least one in 1020 We need Density Estimators that are less prone to overfitting Is there a better way? The problem with the JPT is that it just mirrors the training data. In fact, it is just another way of storing the data: we could reconstruct the original dataset perfectly from it! We need to represent the probability function with fewer parameters… Aside: Bayes Nets Bayes Nets What are they? Bayesian nets are a framework for representing and analyzing models involving uncertainty What are they used for? Intelligent decision aids, data fusion, 3-E feature recognition, intelligent diagnostic aids , automated free text understanding, data mining How are they different from other knowledge representation and probabilistic analysis tools? Uncertainty is handled in a mathematically rigorous yet efficient and simple way Bayes Net Concepts Chain Rule Conditional Independence A Simple Bayes Net Let’s assume that we already have P(Mpg,Horse) How would you rewrite this using the Chain rule? P(good, low) = 0.36 P(good,high) = 0.04 P( bad, low) = 0.12 P( bad,high) = 0.48 Review: Chain Rule P(good, low) = 0.36 P(good,high) = 0.04 P( bad, low) = 0.12 P( bad,high) = 0.48 P(good) = 0.4 P( bad) = 0.6 P( low|good) = 0.89 P( low| bad) = 0.21 P(high|good) = 0.11 P(high| bad) = 0.79 Review: Chain Rule P(good, low) = 0.36 P(good,high) = 0.04 P( bad, low) = 0.12 P( bad,high) = 0.48 P(good) = 0.4 P( bad) = 0.6 P( low|good) = 0.89 P( low| bad) = 0.21 P(high|good) = 0.11 P(high| bad) = 0.79 = P(good) * P(low|good) = 0.4 * 0.89 = P(good) * P(high|good) = 0.4 * 0.11 = P(bad) * P(low|bad) = 0.6 * 0.21 = P(bad) * P(high|bad) = 0.6 * 0.79 P(Mpg, Horse) = P(Mpg) * P(Horse | Mpg) How to Make a Bayes Net P(Mpg, Horse) = P(Mpg) * P(Horse | Mpg) P(good) = 0.4 P( bad) = 0.6 P( low|good) = 0.90 P( low| bad) = 0.21 P(high|good) = 0.10 P(high| bad) = 0.79 How to Make a Bayes Net P(good) = 0.4 P( bad) = 0.6 P( low|good) = 0.90 P( low| bad) = 0.21 P(high|good) = 0.10 P(high| bad) = 0.79 Each node is a probability function Each arc denotes conditional dependence How to Make a Bayes Net So, what have we accomplished thus far? Nothing; we’ve just “Bayes Net-ified” the P(Mpg, Horse) JPT using the Chain rule. …the real excitement starts when we wield conditional independence How to Make a Bayes Net Before we continue, we need a worthier opponent than puny P(Mpg, Horse)… We’ll use P(Mpg, Horse, Accel): P(good, low,slow) = 0.37 P(good, low,fast) = 0.01 P(good,high,slow) = 0.02 P(good,high,fast) = 0.00 P( bad, low,slow) = 0.10 P( bad, low,fast) = 0.12 P( bad,high,slow) = 0.16 P( bad,high,fast) = 0.22 How to Make a Bayes Net Step 1 : Rewrite joint using the Chain rule. P(Mpg, Horse, Accel) = P(Mpg) P(Horse | Mpg) P(Accel | Mpg, Horse) Note: Obviously, we could have written this 3!=6 different ways… P(M, H, A) = P(M) * P(H|M) * P(A|M,H) = P(M) * P(A|M) * P(H|M,A) = P(H) * P(M|H) * P(A|H,M) = P(H) * P(A|H) * P(M|H,A) = … = … How to Make a Bayes Net Step 1 : Rewrite joint using the Chain rule. P(Mpg, Horse, Accel) = P(Mpg) P(Horse | Mpg) P(Accel | Mpg, Horse) How to Make a Bayes Net How to Make a Bayes Net P(good) = 0.4 P( bad) = 0.6 P( low|good) = 0.90 P( low| bad) = 0.21 P(high|good) = 0.10 P(high| bad) = 0.79 P(slow|good, low) = 0.97 P(slow|good,high) = 0.15 P(slow| bad, low) = 0.90 P(slow| bad,high) = 0.05 P(fast|good, low) = 0.03 P(fast|good,high) = 0.85 P(fast| bad, low) = 0.10 P(fast| bad,high) = 0.95 * Note: I made these up too… How to Make a Bayes Net P(good) = 0.4 P( bad) = 0.6 P( low|good) = 0.89 P( low| bad) = 0.21 P(high|good) = 0.11 P(high| bad) = 0.79 P(slow|good, low) = 0.97 P(slow|good,high) = 0.15 P(slow| bad, low) = 0.90 P(slow| bad,high) = 0.05 P(fast|good, low) = 0.03 P(fast|good,high) = 0.85 P(fast| bad, low) = 0.10 P(fast| bad,high) = 0.95 A Miracle Occurs! You are told by God (or another domain expert) that Accel is independent of Mpg given Horse ! i.e., P(Accel | Mpg, Horse) = P(Accel | Horse) How to Make a Bayes Net P(good) = 0.4 P( bad) = 0.6 P( low|good) = 0.89 P( low| bad) = 0.21 P(high|good) = 0.11 P(high| bad) = 0.79 P(slow| low) = 0.22 P(slow|high) = 0.64 P(fast| low) = 0.78 P(fast|high) = 0.36 How to Make a Bayes Net P(good) = 0.4 P( bad) = 0.6 P( low|good) = 0.89 P( low| bad) = 0.21 P(high|good) = 0.11 P(high| bad) = 0.79 P(slow| low) = 0.22 P(slow|high) = 0.64 P(fast| low) = 0.78 P(fast|high) = 0.36 Thank you, domain expert! Now I only need to learn 5 parameters instead of 7 from my data! My parameter estimates will be more accurate as a result! Independence “The Acceleration does not depend on the Mpg once I know the Horsepower.” This can be specified very simply: P(Accel Mpg, Horse) = P(Accel | Horse) This is a powerful statement! It required extra domain knowledge. A different kind of knowledge than numerical probabilities. It needed an understanding of causation. Bayes Nets Formalized A Bayes net (also called a belief network) is an augmented directed acyclic graph, represented by the pair V , E where: V is a set of vertices. E is a set of directed edges joining vertices. No loops of any length are allowed. Each vertex in V contains the following information: A Conditional Probability Table (CPT) indicating how this variable’s probabilities depend on all possible combinations of parental values. Bayes Nets Summary Bayes nets are a factorization of the full JPT which uses the chain rule and conditional independence. They can do everything a JPT can do (like quick, cheap lookups of probabilities) The good news We can do inference. We can compute any conditional probability: P( Some variable Some other variable values ) The good news We can do inference. We can compute any conditional probability: P( Some variable Some other variable values ) Suppose you have m binary-valued variables in your Bayes Net and expression E2 mentions k variables. How much work is the above computation? The sad, bad news Doing inference “JPT-style” by enumerating all matching entries in the joint are expensive: Exponential in the number of variables. But perhaps there are faster ways of querying Bayes nets? In fact, if I ever ask you to manually do a Bayes Net inference, you’ll find there are often many tricks to save you time. So we’ve just got to program our computer to do those tricks too, right? Sadder and worse news: General querying of Bayes nets is NP-complete. Case Study I Pathfinder system. (Heckerman 1991, Probabilistic Similarity Networks, MIT Press, Cambridge MA). Diagnostic system for lymph-node diseases. 60 diseases and 100 symptoms and test-results. 14,000 probabilities Expert consulted to make net. 8 hours to determine variables. 35 hours for net topology. 40 hours for probability table values. Apparently, the experts found it quite easy to invent the causal links and probabilities. Pathfinder is now outperforming the world experts in diagnosis. Being extended to several dozen other medical domains. Bayes Net Info GUI Packages: Genie -- Free Netica -- $$ Hugin -- $$ Non-GUI Packages: All of the above have APIs BNT for MATLAB AUTON code (learning extremely large networks of tens of thousands of nodes) Bayes Nets and Machine Learning Machine Learning Tasks What is an Anomaly? An irregularity that cannot be explained by simple domain models and knowledge Anomaly detection only needs to learn from examples of “normal” system behavior. Classification, on the other hand, would need examples labeled “normal” and “not-normal” Anomaly Detection in Practice Monitoring computer networks for attacks. Monitoring population-wide health data for outbreaks or attacks. Looking for suspicious activity in bank transactions Detecting unusual eBay selling/buying behavior . JPT Anomaly Detector Suppose we have the following model: P(good, low) = 0.36 P(good,high) = 0.04 P( bad, low) = 0.12 P( bad,high) = 0.48 We’re trying to detect anomalous cars. If the next example we see is , how anomalous is it? JPT Anomaly Detector P(good, low) = 0.36 P(good,high) = 0.04 P( bad, low) = 0.12 P( bad,high) = 0.48 Could not be easier! Just look up the entry in the JPT! Smaller numbers are more anomalous in that the model is more surprised to see them. Bayes Net Anomaly Detector P(good) = 0.4 P( bad) = 0.6 P( low|good) = 0.90 P( low| bad) = 0.21 P(high|good) = 0.10 P(high| bad) = 0.79 Bayes Net Anomaly Detector P(good) = 0.4 P( bad) = 0.6 P( low|good) = 0.90 P( low| bad) = 0.21 P(high|good) = 0.10 P(high| bad) = 0.79 Again, trivial! We need to do one tiny lookup for each variable in the network! Machine Learning Tasks Bayes Classifiers Bayes Classifiers in 1 Slide Bayes classifiers just do inference. That’s it. The “algorithm” Learn P(class,X) For a given input x , infer P(class|x ) Choose the class with the highest probability JPT Bayes Classifier Suppose we have the following model: P(good, low) = 0.36 P(good,high) = 0.04 P( bad, low) = 0.12 P( bad,high) = 0.48 We’re trying to classify cars as Mpg = “good” or “bad” If the next example we see is Horse = “low”, how do we classify it? JPT Bayes Classifier P(good, low) = 0.36 P(good,high) = 0.04 P( bad, low) = 0.12 P( bad,high) = 0.48 The P(good | low) = 0.75, so we classify the example as “good” Bayes Net Classifier P(good) = 0.4 P( bad) = 0.6 P( low|good) = 0.89 P( low| bad) = 0.21 P(high|good) = 0.11 P(high| bad) = 0.79 P(slow| low) = 0.95 P(slow|high) = 0.11 P(fast| low) = 0.05 P(fast|high) = 0.89 We’re trying to classify cars as Mpg = “good” or “bad” If the next example we see is how do we classify it? Suppose we get a example? P(good) = 0.4 P( bad) = 0.6 P( low|good) = 0.89 P( low| bad) = 0.21 P(high|good) = 0.11 P(high| bad) = 0.79 P(slow| low) = 0.95 P(slow|high) = 0.11 P(fast| low) = 0.05 P(fast|high) = 0.89 Note: this is not exactly 0.75 because I rounded some of the CPT numbers earlier… Bayes Net Bayes Classifier P(good) = 0.4 P( bad) = 0.6 P( low|good) = 0.89 P( low| bad) = 0.21 P(high|good) = 0.11 P(high| bad) = 0.79 P(slow| low) = 0.95 P(slow|high) = 0.11 P(fast| low) = 0.05 P(fast|high) = 0.89 The P(good | low, fast) = 0.75, so we classify the example as “good”. …but that seems somehow familiar… Wasn’t that the same answer as P(Mpg=good | Horse=low)? Bayes Classifiers OK, so classification can be posed as inference In fact, virtually all machine learning tasks are a form of inference Anomaly detection: P(x ) Classification: P(Class | x ) Regression: P(Y | x ) Model Learning: P(Model | dataset) Feature Selection: P(Model | dataset) The Naïve Bayes Classifier ASSUMPTION: all the attributes are conditionally independent given the class variable The Naïve Bayes Advantage At least 256 parameters! You better have the data to support them… A mere 25 parameters! (the CPTs are tiny because the attribute nodes only have one parent.) What is the Probability Function of the Naïve Bayes? P(Mpg,Cylinders,Weight,Maker,…) = P(Mpg) P(Cylinders|Mpg) P(Weight|Mpg) P(Maker|Mpg) … What is the Probability Function of the Naïve Bayes? Bayes Classifier Results: “MPG”: 392 records The Classifier learned by “Naive BC” Bayes Classifier Results: “MPG”: 40 records More Facts About Bayes Classifiers Many other density estimators can be slotted in Density estimation can be performed with real-valued inputs Bayes Classifiers can be built with real-valued inputs Rather Technical Complaint: Bayes Classifiers don’t try to be maximally discriminative---they merely try to honestly model what’s going on Zero probabilities are painful for Joint and Naïve. A hack (justifiable with the magic words “Dirichlet Prior”) can help. Naïve Bayes is wonderfully cheap. And survives 10,000 attributes cheerfully! Summary Axioms of Probability Bayes nets are created by chain rule conditional independence Bayes Nets can do Inference Anomaly Detection Classification The Axioms Of Probability
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